(x-3)^2 + (y-2)^2 = 16

3 min read Jun 17, 2024

Understanding the Equation: (x - 3)^2 + (y - 2)^2 = 16

The equation (x - 3)^2 + (y - 2)^2 = 16 represents a circle in the coordinate plane. Let's break down why and how to interpret this equation.

The Standard Equation of a Circle

The standard form of the equation of a circle is:

(x - h)^2 + (y - k)^2 = r^2

where:

(h, k) represents the center of the circle.
r represents the radius of the circle.

Applying the Equation to Our Example

Comparing our given equation, (x - 3)^2 + (y - 2)^2 = 16, to the standard form, we can identify the following:

Center: (h, k) = (3, 2)
Radius: r^2 = 16, so r = 4

Graphing the Circle

To graph the circle, follow these steps:

Plot the center: Mark the point (3, 2) on the coordinate plane.
Draw the radius: From the center, draw a line segment of length 4 units in each of the four cardinal directions (up, down, left, right).
Complete the circle: Connect the endpoints of these line segments to form a smooth circle.

Key Properties

Here are some key properties of the circle represented by the equation (x - 3)^2 + (y - 2)^2 = 16:

Center: (3, 2)
Radius: 4
Diameter: 8
Circumference: 2πr = 8π
Area: πr^2 = 16π

Applications

The equation of a circle has various applications in different fields, including:

Geometry: Analyzing shapes and their properties.
Physics: Modeling the motion of objects in circular paths.
Engineering: Designing circular structures and components.

Understanding the equation of a circle is fundamental to many areas of mathematics and science. By applying this knowledge, we can analyze, interpret, and manipulate circular shapes effectively.